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Addition and Subtraction

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Outline Arithmetic Operations (Section 1.2) – Addition – Subtraction – Multiplication Complements (Section 1.5) – 1’s complement – 2’s complement Signed Binary Numbers (Section 1.6) – 2’s complement – Addition – Subtraction

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Addition 101101+100111 Rules – Any carry obtained in a given significant position is used by the pair of digits one significant position higher

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Subtraction 101101-100111 The borrow in a given significant position adds 2 to a minuend digit

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Multiplication 1011 X 101

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Complement 1’s complement 2’s complement

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1’s complement Rule: 1’s complement of a binary number is formed by changing – 1’s to 0’s – 0’s to 1’s 1011000→0100111 0101101 →

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2’s Complement Alternative Method – Write the 1’s complement – Add 000…1 to 1’s complement Example – 1101100 – 0010011 (1’s complement) – 0010100 (2’s complement)

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Unsigned Subtraction X-Y – Determine Y’s 2’s complement – X+(2’s complement of Y) If X is larger or equal to Y, an end carry will result. Discard the end carry. If X is less than Y, no end carry will result. To obtain the answer in a familiar form, take the 2’s complement of the sum and place a negative sign in front.

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Subtraction of Unsigned Number Example 1.7 (2’s complement) X=1010100 Y=1000011 X-Y (Discard end carry) Y-X (No end carry)

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Signed Binary Number Signed-magnitude representation Signed 1’s complement representation Signed 2’s complement representation

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Signed Magnitude Representation Rules – Represent the sign in the leftmost position – 0 for positive – 1 for negative Example – 01001↔(+)9 – 11001↔(-)9 Used in ordinary arithmetic, but not in computer arithmetic

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Interpretation The user determines whether the number is signed or unsigned – 01001 9 (unsigned binary) +9 (signed binary) – 11001 25 (unsigned binary) -9 (signed binary)

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Signed Complement System A signed complement system negates a number by taking its complement Example – 00001001 (9) – 11110110 (-)9 in signed 1’s complement – 11110111 (-)9 in signed 2’s complement Usage: – 1’s complement: Seldom used – 2’s complement: Most common

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Arithmetic Addition The rule for adding signed numbers in 2’s complement form is obtained from addition of two numbers A carry out of the sign bit is discarded In order to obtain correct answer, we must ensure that the result has a sufficient number of bits to accommodate the sum Useful Facts – Positive numbers have 0 in the leftmost bit – Negative numbers have a 1 in the leftmost bit

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Negative Number Determine the value of a negative number in signed 2’s complement by converting the number to a positive number to place it in a more familiar form – 11111001 is negative because the left most bit is 1. – 2’s complement: 00000111 (+7) – Therefore, 11111001 is -7

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